15.4.15 problem 16

Internal problem ID [2928]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 8, page 34
Problem number : 16
Date solved : Sunday, March 30, 2025 at 12:56:40 AM
CAS classification : [_exact, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{2} \csc \left (x \right )^{2}+6 x y-2&=\left (2 y \cot \left (x \right )-3 x^{2}\right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 65

ode:=y(x)^2*csc(x)^2+6*x*y(x)-2 = (2*y(x)*cot(x)-3*x^2)*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {3 x^{2} \tan \left (x \right )}{2}-\frac {\sqrt {\tan \left (x \right ) \left (9 \tan \left (x \right ) x^{4}+4 c_1 -8 x \right )}}{2} \\ y &= \frac {3 x^{2} \tan \left (x \right )}{2}+\frac {\sqrt {\tan \left (x \right ) \left (9 \tan \left (x \right ) x^{4}+4 c_1 -8 x \right )}}{2} \\ \end{align*}

✓ Mathematica. Time used: 32.781 (sec). Leaf size: 201

ode=y[x]^2*Csc[x]^2+6*x*y[x]-2==(2*y[x]*Cot[x]-3*x^2)*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {3}{2} x^2 \tan (x)-\frac {\csc (2 x) \sqrt {-\left (\tan (x) \left (16 \cos ^2(x) \arcsin \left (\sqrt {\sin ^2(x)}\right )-9 x^4 e^{\text {arctanh}(\cos (2 x))}+\cos (2 x) \left (9 x^4 e^{\text {arctanh}(\cos (2 x))}-4 c_1\right )-4 c_1\right )\right )}}{2 \sqrt {\csc (2 x) e^{\text {arctanh}(\cos (2 x))}}} \\ y(x)\to \frac {3}{2} x^2 \tan (x)+\frac {\csc (2 x) \sqrt {-\left (\tan (x) \left (16 \cos ^2(x) \arcsin \left (\sqrt {\sin ^2(x)}\right )-9 x^4 e^{\text {arctanh}(\cos (2 x))}+\cos (2 x) \left (9 x^4 e^{\text {arctanh}(\cos (2 x))}-4 c_1\right )-4 c_1\right )\right )}}{2 \sqrt {\csc (2 x) e^{\text {arctanh}(\cos (2 x))}}} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*x*y(x) - (-3*x**2 + 2*y(x)/tan(x))*Derivative(y(x), x) + y(x)**2/sin(x)**2 - 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out